124 ANALYTICAL MECHANICS Thus for a given value of ft R1 is maximum when sin (2 a - /? = 1, that is, when a = ^ + -: = _ _ _ ^ cos2/5 0(l+.sini3) When 0 = 0 equations (15) and (16) reduce to equations (12) and (13), as they should. PROBLEMS. 1. A body is projected horizontally with a speed of 105 — '- from a cliff sec. 365 feet high. Find the magnitude and direction of the velocity at the time it reaches the ground. 2. The muzzle velocity of a gun is 3000 — - . Find the area it covers sec. if it is mounted on top of a hill 500 feet above the surrounding plain. 3. A shot fired horizontally from the top of a tower strikes the ground at a distance d from the base of the tower, with a velocity the vertical component of which equals the initial velocity of the shot. Find the height of the tower. 4. A bullet is projected at an angular elevation of 45° with a velocity of 400 — . At the highest point of its flight the bullet goes through a sec. target 5 cm. thick and strikes the ground at a distance of 1200 m. from . the place where it was projected. Find the average resisting force offered by the target. 6. After sliding 200 m. down a slope of 30° a ski-jumper leaves the ground making 45° with the horizon and lands further down the same sloping ground. Supposing the coefficient of friction to be 0.05 and neglecting the resistance due to air, find (a) the speed with which he left the ground, (b) the speed with which he landed. 6. In the preceding problem find the leap measured along the ground. 7. A man can make 6 feet in the high jump. How many feet could he make if he were on the moon? The gravitational acceleration on the moon - 0 ft. is 5.3 — - . sec.2 8. A particle slides down a chord of a vertical circle and then falls on a horizontal plane h feet below the lower end of the chord. Find the chord which will give the greatest possible range on the plane.